A recently developed nonlocal vector calculus is exploited to provide a variational analysis
for a general class of nonlocal diffusion problems described by a linear integral equation on
bounded domains in Rn. The nonlocal vector calculus also enables striking analogies to be
drawn between the nonlocal model and classical models for diffusion, including a notion
of nonlocal flux. The ubiquity of the nonlocal operator in applications is illustrated by a
number of examples ranging from continuum mechanics to graph theory. In particular, it is
shown that fractional Laplacian and fractional derivative models for anomalous diffusion
are special cases of the nonlocal model for diffusion that we consider. The numerous
applications elucidate different interpretations of the operator and the associated governing
equations. For example, a probabilistic perspective explains that the nonlocal spatial
operator appearing in our model corresponds to the infinitesimal generator for a symmetric
jump process. Sufficient conditions on the kernel of the nonlocal operator and the notion
of volume constraints are shown to lead to a well-posed problem. Volume constraints are
a proxy for boundary conditions that may not be defined for a given kernel. In particular,
we demonstrate for a general class of kernels that the nonlocal operator is a mapping
between a volume constrained subspace of a fractional Sobolev subspace and its dual.
We also demonstrate for other particular kernels that the inverse of the operator does
not smooth but does correspond to diffusion. The impact of our results is that both a
continuum analysis and a numerical method for the modeling of anomalous diffusion on
bounded domains in Rn are provided. The analytical framework allows us to consider
finite-dimensional approximations using discontinuous and continuous Galerkin methods,
both of which are conforming for the nonlocal diffusion equation we consider; error and
condition number estimates are derived.